A geometric classification of some solvable groups of homeomorphisms
Collin Bleak
备注：
19 pages, in preparation. Portions of this paper contain portions of the author's dissertation written at Binghamton University. Newer versions contain improved grammar, corrected typos and minor re-arrangements
We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of the generalized Thompson groups F_n. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product C wr T (of non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphic with the integers.
We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups. We prove that, if $K$ is a perfect field and $X$ is a homogeneous space of a smooth algebraic $K$-group $G$ with finite geometric stabilizers lying in this family, then $X$ is dominated by a $G$-torsor. In particular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rational points.
The Word and Geodesic Problems in Free Solvable Groups
研究了自由可解群$S_{r,d}$上的Word问题的计算复杂性，其中$r \geq 2$是群的秩，$d \geq 2$是群的可解类.已知将$S_{r,d}$嵌入到矩阵中的Magnus嵌入为固定群$S_{r,d}$中的WP提供了多项式时间判决算法。但由于多项式的次数随着$d$的增长而增长，所以在$d$中统一算法不是多项式的.在本文中，我们证明了WP的时间复杂度$O(r n \log_2 n)$在$S_{r,2}$，和$O(n^3 r d)$在$S_{r,d}$为$d \geq 3$。然而，事实证明，计算$S_{r,2}$中元素的测地线长度的看似接近的问题是$NP$ -完全的。我们还证明，人们可以计算福克斯衍生物的元素从$S_{r,d}$在时间$O(n^3 r d)$，特别是可以有效地使用Magnus嵌入计算与自由可解群。我们的方法是基于这样的经典工具，如Magnus嵌入和福克斯演算，以及，在一个相对较新的几何思想，特别是，我们建立了一个直接的联系福克斯衍生物和凯莱图上的几何流。
Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$…
